"Ordinary" toric varieties over the complex numbers exhibit many of the behaviors of more complicated varieties but are computationally manageable. Arithmetic toric varieties play a similar role for questions over non-closed fields. In particular, while toric varieties can be seen as generalizations of projective spaces, arithmetic toric varieties can be seen as generalizations of Severi-Brauer varieties. The endomorphism algebras of exceptional objects in the derived category of an arithmetic toric variety are one way of extending the connection between Severi-Brauer varieties and central simple algebras. We discuss how properties of these algebras succeed and fail to reflect rationality properties of the varieties.
This is based on joint work with Matthew Ballard, Alicia Lamarche, and Patrick McFaddin.